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Number theory in the spirit of Liouville. (English) Zbl 1227.11002
London Mathematical Society Student Texts 76. Cambridge: Cambridge University Press (ISBN 978-0-521-17562-3/pbk; 978-1-107-00253-1/hbk). xvii, 287 p. £ 29.99/pbk, $49.00/pbk; £ 70.00,$ 115.00/hbk (2011).
The book starts with a brief biography of the French mathematician Joseph Liouville. In a series of eighteen papers published during the years 1858 and 1865 Liouville introduced a powerful new method into elementary number theory. His idea was to give a number of elementary identities from which many number-theoretic results originated by specializing the functions involved in the formulae. In this book a gentle introduction to Liouville’s method has been provided. A sufficient number of his identities has been given in order to provide elementary arithmetic proofs of such number-theoretic results as the Girard-Fermat theorem, a recurrence relation for the sum of divisors function, Lagrange’s theorem, Legendre’s formula for the number of representations of a nonnegative integer as the sum of four triangular numbers, Jacobi’s formula for the number of representations of a positive integer as the sum of eight squares, and many others. Some of the more recent results that have been obtained using Liouville’s ideas have also been given in this book. Liouville’s ideas have been used to give elementary proofs of many arithmetic formulae. Using more advanced mathematics, such as the theory of modular forms, one can prove these and other formulae. In the last chapter of the book a sketch has been given to use the theory of modular forms to prove arithmetic identities.

##### MSC:
 11-01 Textbooks (number theory) 11Axx Elementary number theory 11E25 Sums of squares, etc 11F27 Theta series; Weil representation; theta correspondences